Stress-Energy tensor for dust

Summary:: I find two different expressions for the EM tensor for dust, and both derivations seem right to me.

Given the action for a system of dust $$S =-sum m_q int sqrt{g_{munu}[x_q(lambda)]dot{x}^mu_q(lambda)dot{x}^nu_q(lambda)} dlambda,$$
where I use the $(+,-,-,-)$ sign convention.
The Energy-Momentum Tensor (EMT) is defined by the variation of the metric

$$delta S = frac{1}{2}int T_{munu} delta g^{munu} sqrt{g} d^4x.$$

To compute that I use two different approaches, first one, because I want to vary $g^{munu}$ I find it better to write $S =-sum m_q int sqrt{g^{munu}[x_q(lambda)]dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)} dlambda$. Then

$$delta S = -sum m_q int frac{dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}{2sqrt{g^{munu}[x_q(lambda)]dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}} delta g^{munu}dlambda.$$

And multiplying by $1=int delta^{(4)}(x^mu – x^{mu}_q(lambda))frac{sqrt{g}}{sqrt{g}} d^4x$

$$delta S = -frac{1}{2}sum m_q int frac{delta^{(4)}(x^mu – x^{mu}_q(lambda))dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}{sqrt{g}sqrt{g^{munu}[x_q(lambda)]dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}} delta g^{munu}dlambda sqrt{g}d^4x.$$

Giving

$$T_{munu} = -sum m_q int frac{delta^{(4)}(x^mu – x^{mu}_q(lambda))dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}{sqrt{g}sqrt{g^{munu}[x_q(lambda)]dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}} dlambda.$$

The second approach, is by doing the variation to $g_{munu}$, doing exactly the same steps I get

$$delta S = -frac{1}{2}sum m_q int frac{delta^{(4)}(x^mu – x^{mu}_q(lambda))dot{x}^mu_{q}(lambda)dot{x}_{q}^nu(lambda)}{sqrt{g}sqrt{g_{munu}[x_q(lambda)]dot{x}_{q}^mu(lambda)dot{x}_{q}^nu(lambda)}} delta g_{munu}dlambda sqrt{g}d^4x.$$

Now, because $0=delta(g_{munu}g^{nulambda})$ we must have $delta g_{munu} = -g_{mualpha}g_{nubeta}delta g^{alphabeta}$ so I find

$$delta S = frac{1}{2}sum m_q int frac{delta^{(4)}(x^mu – x^{mu}_q(lambda))dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}{sqrt{g}sqrt{g^{munu}[x_q(lambda)]dot{x}_{qmu}(lambda)dot{x}_{qnu}(lambda)}} delta g^{munu}dlambda sqrt{g}d^4x.$$

Giving an EMT equal, but with a negative sign. The second one seems better because gives an energy density bounded for below, while the first one not, but I don’t see any mistake. Furthermore, because the two derivations are so similar, I don’t think an algebraic mistake can explain such difference, so the error must be a conceptual one.